Section B.1 Problem Sheet 1
Due 4 October at 3pm, either in class or in the pigeon hole in 4W.
Exercise B.1. Quantifiers and limits.
Let \(\seq an\) be a sequence of real numbers and let \(L \in \R\text{.}\)
(a)
Write the precise definition of “\(a_n \to L\) as \(n\to \infty\)”. Spell out any quantifiers \(\forall,\exists\) as words.
(b)
Which of the following is the logical negation of the statement “\(a_n \to L\) as \(n \to \infty\)”?
For all \(\varepsilon \gt 0\text{,}\) and for all \(N \in \N\text{,}\) there exists \(n \ge N\) such that \(\abs{a_n-L} \ge \varepsilon\text{.}\)
There exists \(\varepsilon \gt 0\) such that, for all \(N \in \N\text{,}\) there exists \(n \ge N\) such that \(\abs{a_n-L} \ge \varepsilon\text{.}\)
There exists \(\varepsilon \gt 0\) and \(N \in \N\) such that, for all \(n \ge N\text{,}\) \(\abs{a_n-L} \ge \varepsilon\text{.}\)
For all \(\varepsilon \gt 0\text{,}\) there exists \(N \in \N\) such that, for all \(n \ge N\text{,}\) \(\abs{a_n-L} \ge \varepsilon\text{.}\)
(c)
Let \(\seq an\) and \(\seq bn\) be sequences of real numbers with \(\lim_{n \to \infty} a_n = 0\) and \(\lim_{n \to \infty} b_n = 1\text{.}\) The conclusion of the argument below is clearly incorrect, but where does the reasoning go wrong? How could the argument have been written to make such a mistake less likely? Note that I am not asking you to give a correct proof of a related true statement.
For any \(\varepsilon_1 \gt 0\text{,}\) we can find \(N_1 \in \N\) such that \(\abs{a_n} \lt \varepsilon_1\) for all \(n \ge N_1\text{.}\) Similarly, for any \(\varepsilon_2 \gt 0\) we can find \(N_2 \in \N\) such that \(\abs{b_n-1} \le \varepsilon_2\) for \(n \ge
N_2\text{.}\) Let \(N=\max\{N_1,N_2\}\text{.}\) Then for \(n \ge N\text{,}\)
\begin{align*}
\abs{a_n+b_n}
\amp = \abs{a_n+(b_n-1)+1}\\
\amp \le \abs{a_n}+\abs{b_n-1}+1\\
\amp \lt \varepsilon_1 + \varepsilon_2 + 1\\
\amp = \varepsilon\text{.}
\end{align*}
Therefore \(\lim_{n\to \infty}(a_n + b_n)=0\text{.}\)
Exercise B.2. Suprema and limits.
Let \(A \subset \R\) be a nonempty set.
(a)
Write down the definition of \(\sup A\) in complete detail.
Hint.
If you have any doubts, look it up in your old notes!
(b)
Suppose that \(\sup A \lt \infty\text{.}\) Using the above definition, show that there exists a sequence \(\seq an\) of points in \(A\) such that \(a_n \to \sup A\) as \(n \to \infty\text{.}\)
Hint.
Let \(n \in \N\text{.}\) Is \(\sup A - 1/n\) an upper bound for \(A\text{?}\)
(c)
Give counterexamples to show that each of the arguments below is incorrect.
Suppose that every \(a \in A\) satisfies \(a \lt \infty\text{.}\) Then \(\sup A
\lt \infty\text{.}\)
Suppose that every \(a \in A\) satisfies \(a \lt 1\text{.}\) Then \(\sup A \lt
1\text{.}\)
Hint.
Both arguments are valid when \(A\) is a finite set, in which case \(\sup\) can be replaced by \(\max\text{.}\) To find counterexamples, you will therefore need to think about infinite sets \(A\text{.}\)
Exercise B.3. Inequalities and limits.
Let \(\seq an\text{,}\) \(\seq bn\) be convergent sequences of real numbers, with \(a_n \to a\) and \(b_n \to b\) as \(n \to \infty\text{.}\)
(a)
Directly using the definition of the limit of a sequence, show that if \(a \lt b\text{,}\) then there exists \(N \in \N\) such that \(a_n \lt b_n\) for all \(n \ge N\text{.}\)
Hint.
Try sketching the sequences and their limits on a number line to get inspiration.
(b)
Find an example where \(a \le b\text{,}\) but \(a_n \gt b_n\) for all \(n
\in \N\text{.}\)
(c)
Directly using the definition of the limit of a sequence, show that if \(a_n
\le b_n\) for all \(n \in \N\text{,}\) then \(a \le b\text{.}\)
Hint.
Try proving the contrapositive.
(d)
Find an example where \(a_n \lt b_n\) for all \(n \in \N\) but \(a \ge
b\text{.}\)
While
Chapter 1 begins with a review of Analysis 2A, my experience has been that many students would also benefit from a review of Analysis 1. So, if you are feeling
at all rusty with Analysis 1 techniques, I encourage you in the strongest possible terms to attempt some of the questions on this problem sheet and turn them in so that you can get feedback.
Please feel free to email me, or drop by office hours, which in Week 1 are Wednesday 10:15–11:05 in 4W 1.12, with any questions about these problems whatsoever. I am also more than happy to meet one-on-one or in a small group.
In particular, if these problems are very difficult for you, please get in touch with me over email or in office hours as soon as you can. It is possible that you will need to do a more thorough revision of Analysis 1 and/or that this unit may not be a great fit for you.
